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Discrete Dynamics in Nature and Society, Vol. 5, pp. 203-221
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Bifurcation Analysis of the Henon Map
ZHANYBAI T. ZHUSUBALIYEV ’*, VADIM N. RUDAKOV a,
EVGENIY A. SOUKHOTERIN and ERIK MOSEKILDE b
aKursk State Technical University, Department of Computer Science, 50 Years of October Street,
94, 305040 Kursk, Russia," b Center for Chaos and Turbulence Studies, Department of Physics,
Technical University of Denmark, 2800 Lyngby, Denmark
(Received 15 March 2000)
Division of the parameter plane for the two-dimensional H6non mapping into domains
of periodic and chaotic oscillations is studied numerically and analytically. Regularities
in the occurrence of different motions and transitions are analyzed. It is shown that
there are domains in the plane of parameters, where non-uniqueness of motions exists.
This may lead to abrupt changes of the character of the dynamics under variation in the
parameters, that is, to a sudden transition from one stable cycle to another or to
chaotization of the oscillations.
Keywords: H6non two-dimensional mapping; Chaos; Bifurcations; Branching pattern
Here a is the nonlinearity coefficient and/3 is the
coefficient of dissipation ([/31 < 1). T denotes the
operation of transposition.
The mapping (1) was originally proposed by
H6non (Hdnon, 1976) as a model of the Poincar6
map for a three-dimensional flow system. Since
1. INTRODUCTION
The purpose of this paper is to apply bifurcation
analysis and continuation methods to study regularities in the occurrence of different motions in
the two-dimensional H6non map (H6non, 1976;
Hitzl and Zele, 1981):
then it has been extensively studied both numerically and from a more theoretical point of view
(for references see, e.g., Schuster, 1984; Berg6 et al.,
1984; Moon, 1987; Anishchenko, 1990; Landa,
1996). Curry (Curry, 1979) was the first to observe
that there exists a range of parameters for which
two chaotic attractors coexist, and Derrida et al.
(Derrida et al., 1979) showed how the
Xk--F(Xk-1); k--l,2,...;
X- (xl, x2) r" F- (fl,f2) r"
fl
OZX -+- X2;
*Corresponding author.
203
204
Z.T. ZHUSUBALIYEV et al.
period-doubling cascade in the H6non map for certain parameter values follows the Feigenbaum
scenario. At the same time, Newhouse (Newhouse,
1974) found that there are regions in parameter
space in which a dense set of the stable periodic orbits with extremely small basins of attraction may
exist. For a more recent discussion of this phenomenon see, e.g., Kan et al. (Kan et al., 1995). A variety of different motions generated by the
two-dimensional H6non map (Simo, 1979; Hansen
and Cvitanovi6, 1998) in dependence of the parameters and peculiarities in the occurrence of chaos
are typical of a wide class of dynamical systems
(Schuster, 1984; Moon, 1987; Anishchenko, 1990;
Mosekilde, 1996). As shown, for instance, by
Anishchenko (Anishchenko, 1990) the map (1) is a
good representation of a three-dimensional model
of a radiophysical generator with inertial nonlinearity. In particular, it is shown that the collection of bifurcations in the plane of the controlling
parameters c and/3 is qualitatively equivalent to
that which is realized in the Poincar6 map of a
model of the generator. We have found a similar
equivalence when analyzing models of a specific
class of automatic control systems, for example,
two- and three-dimensional models of systems
with pulse-width modulation and a four-dimensional model of a relay system with hysteresis
(Baushev and Zhusubaliyev, 1992; Baushev et al.,
1996; Zhusubaliyev, 1997; Zhusubaliyev, 1997a).
Many other non-linear dynamical systems display a structure of division of the parameter plane
and a shape and arrangement of the domains of
stability similar to what we shall discuss here. The
peculiarities of organization of domains of stability with the specific shape, referred to as a swallow
tail, or a crossroad area and denoted here as
walw can be observed with bimodal one-dimensional maps as shown in (Gallas, 1994), for the
Ikeda map (Mosekilde, 1996), and for several
other nonlinear dynamical systems (Barfred et al.,
1996). Detailed analysis of such a structure have
been performed by Kuznetsov et al. (Kuznetsov
et al., 1993) and, from a more mathematical point
of view, by Mira et al. (Mira et al., 1996) (see also
II
references in this work). It is shown in (Kuznetsov
et al., 1994), that a quantitative universality in the
transition to chaos in two-dimensional H6non-like
maps exists. Hansen and Cvitanovi6 (Hansen and
Cvitanovi6, 1998) describe the possible bifurcation
structures for maps of the H6non type, in particular some swallow tail for various periods using
a series of n-unimodal approximations and the
associated symbolic dynamics. Sonis (Sonis, 1996)
provides a detailed description of the bifurcation
phenomena in the H6non map.
Nonetheless, the complete bifurcation scenario
for the H6non map has not yet been established,
and the aim of the present article is to present a
number of results that appear not to have been
given in the above works. Special emphasis is paid
to the study of the structure and properties of the
division of the plane of controlling parameters for
the dynamical system (1) into the domains of
existence of periodic motions and chaos.
2. DIVISION OF PARAMETER PLANE
Let us begin with some preliminary notes.
Let Xci, i= 1,m be a periodic motion (m-cycle)
of the dynamic system (1). It is evident that all Xc;,
i-- 1, m satisfy the following equation
Xc Fm(Xc)
O, Fm(Xc)
F(F(... (F(Xc)) ...))
times
(2)
The local stability of the m-cycle is determined
by the eigenvalues (multiplicators) pl and p2 of the
These are the roots of the
basic matrix
equation
m.
det(m
pE)
OF(Xc(i-1)) i-l, GO
OXc( iOF(Xc(i-1))
OXc( i-
O,
1,m
E,i
-2OXlc(i-1)
o
BIFURCATIONS OF THE
Here E is the unit matrix.
Let II {(c,/3): # < c < u; </3 < } be the set
of the parameters of the dynamic system (1). A
point in the plane P (c,/3) EII corresponds to
every fixed set of parameter values. Let X.i(P),
1,m be a locally stable m-cycle corresponding to
point P.
Let IIk,j- be the simply connected set of parameters I-[k, j C I-I, k 1; j 1,2,..., s such, that for
every P E IIk,j there exists a stable m-cycle X,i(P),
1, m, which is continuous along the parameters
in IIk,.. If there are several motions at fixed P,
different IIk,v. correspond to different motions. The
k-cycle for Ilkd is minimum, that is, there is no mcycles with m < k in llk,.. The index j is introduced
to distinguish sets, having the same k. The value of
HI’NON MAP
denote these curves as N+ and N_, respectively.
For the dissipative system (1), the case where two
complex conjugate eigenvalues cross the unit circle
cannot occur.
It is interesting to study the possibility of nonempty intersections of the sets Yik,j and the
properties of IIk,v. in whole in order to understand
the great variety of different motions, generated by
the dynamical system (1). In the following we shall
restrain ourselves to the region II={(c,/3):
-1<c<4; 1/31<1}.
Let us first consider the solutions of Eq. (2) and
study the local stability of m-cycles.
has the form:
Equation (2) when m
+
s may be finite or infinite.
X2
The boundaries of the sets
following types:
IIk,.. may be of the
simple, that is, there is a stable m-cycle for any
points P 1-’k,v-, connected with a limitation of
the parameter variation range;
the boundaries Pk,. are formed by the collection
of bifurcation values of parameters or by the
collection of points of accumulation, corresponding to aperiodic motions.
The point P, II is a bifurcation point, if the
equation (Neymark and Landa, 1987; Butenin
et al., 1987)
x(P, P)
x(P, p)
[92 --]-
205
lP @ 2
det(m
0;
Xlc"
-
The roots of this equation determine two
period- cycles
X
X2c
/Xlc.
The domain of existence of a stable 1-cycle is
limited by bifurcation curves N+, N_
/32-2/+4c+1 =0
(4)
and
3/2 6/3
O,
4c + 3
O,
pE)
respectively. Let us denote this domain as 1-I1,1
has a root, which is on the unit circle, when P- P..
These are the cases, when the greatest multiplicator of m-cycle (in absolute value) turns into
or -1. The corresponding bifurcation curves are
determined by the equations
x(P, 1)
x(P,-1)
+1 @2
1 @2
0
0
By analogy with (Neymark and Landa, 1987;
Butenin et al., 1987) here and in the following we
II1, II1 { (OZ,/)
1) 2, I/l < }, then Eq. (3) has
If the parameters P
<c<
(1/4)(/3
no real roots. On the curve c
-1 </3 <
Eq. (3) takes the form
=0
(1/4)(/3- 1)2,
Z.T. ZHUSUBALIYEV
206
and has a multiple root. When changing the
parameters along some smooth curve (referred to
as the trajectory of deformation) into the domain
a>-(1/4)(/3-1)2, -1</3<1, two solutions
arise: one of which corresponds to a stable 1-cycle
and the other to an unstable 1-cycle. This pair of
1-cycles occurs abruptly in a saddle-node bifurcation at a--(1/4)(3-1) 2. In a reverse transition,
the stable 1-cycle disappears as it coincides with the
unstable cycle at the points of intersection with the
trajectory of deformation and curve (4).
It is easy to see, that a- (1/4)(/3- 1)2 < 0.
Now let us follow the evolution of the stable and
unstable 1-cycles with variation in a from negative values to positive ones by passing through
a-0. Denote the solutions of the Eq. (2) as X
XUc, that correspond to stable 1-cycle and unstable
13=0.2
o
ot
(a)
s,
-cycle.
When a
et al.
PlP2
[3=0.2
P;
0, Eq. (2) has only one solution
Xlc
It is obvious, that
’
X2c
3
0.3
-0. t6
lim
-,+/-o
s
x
1-/3’
lim
-++o
xzSc
Hence the solution of Eq. (2), which corresponds to the stable 1-cycle, depends smoothly on
the parameters in IIl,1, whereas XU is uninterrupted in 1-[1,1 except for the points, in which a- 0.
In points a-0, 1/31 <1 the solution XcU has an
infinite discontinuity.
The distinction in the nature of the parameter
dependence of the solutions Xcs and XcU is shown
in Figure l(a). The dependences of the multiplicators of stable and unstable 1-cycles are given in
Figure (b).
When the trajectory of deformation and the
bifurcation curve (5) intersect, the 1-cycle loses its
stability, and a stable 2-cycle arises softly as the
result of a period-doubling bifurcation. Both 1cycles continue to exist as unstable over the whole
range of parameters beyond bifurcation curve (5).
FIGURE
(a) Variation of the stable and unstable period-1
cycles with the nonlinearity parameter c. Note the different
nature of these dependences. (b) Corresponding variation of the
multiplicators Pl and/)2.
Equation (2) when m- 2 takes the form
oz3X41c 2o2Xc q- (1 fl)3Xlc- (1 -/)2 @ o
X2c
1-
0;
This equation has four real roots in the domain
a
> (3/4)(/3- 1) 2. Two of the roots correspond to
two unstable 1-cycles. It is not difficult to show,
that a stable 2-cycle satisfies the equation
O2Xc
X2c
c(1 -/3)Xlc + (1 -/)2
(1 aX21c)"
-/3
o
0;
(6)
BIFURCATIONS OF THE
6/3
4c
5
(7)
0
that is,
H2,-
(oz,/3)’(/3-1
_<oz
<1
5), I/ 1 <
(5/32
6/3
o6Xtc o5(1 t)Xc oz4((1 )2 3oz)Xlc_
nL
}.
o(1 )(1 -t-/32 2oz)Xc+
qL_ 2 (4 2 @
(3 @ 2. ).
+
ty
When passing through the boundary (7) into the
domain of values c > (1/4)(5/32- 6/3-5), 1/31 < 1, a
stable 4-cycle arises softly, and the 2-cycle continues to exist, but becomes a saddle.
It becomes more and more difficult to obtain
bifurcation formulas in explicit form for m- 2;-1,
i- 3, 4,... All the rest of the sets H2/-l,1, i-- 3, 4,...
were built numerically. The collection of H2,-l,1,
i-- 1,2,... which forms 1-I1,1, is shown in Figure 2:
(1
2+2
4fl5fl2 4fls + 4_ 2fl2)
0.
The roots of this equation can be found only
numerically. Let us try to find the equation of
bifurcation curve N+ for the 3-cycle. Taking into
account, that with parameter values, located on
the curve N+, Eq. (8) has only multiple roots, we
U fi>’,,
n,,,
207
In this way, the set II1,1 consists of domains of
existence of stable 2i-l-cycles (i-1,2, 3,...). The
boundaries, separating these domains in
correspond to period-doubling bifurcation curves.
The set I-[1,1 is limited by bifurcation curve (4)
from the left and by a curve formed by the
collection of accumulation points from the right.
Now consider I-[k, j --Uie=l l-Ik.2i-l,j for k-/= 1.
From Eq. (2) when m- 3 we find
The set H2,1, whereupon a stable 2-cycle is
determined, is limited by the bifurcation curve (5)
and by the curve, defined by the following
equation
5/2
HI’NON MAP
i=1
-1
-1
FIGURE 2 The collection of the sets I]2,-,,1,
diverge to infinity with any initial conditions.
et
4
1,2,... which forms 1-I,1, I] and I denote those domains of H, where trajectories
Z.T. ZHUSUBALIYEV et al.
208
obtain
2c
/Xc + x +
which were obtained numerically. Naturally,
Figure 4 represents only those sets, that are
relatively large in II.
The resulting picture for the division of Hdnon
o.
Here
2
-/3-/3 +/3
2/ (2c c
qt_ /4 qt_/5 nt O2/_ 2Ct/33)
o-
where
0
+/3 -+-/2 q__/4 q_ 5 q_/6
c( 2c q- O 2 4/3 5/2
4 +/4 2cfl2).
Then the desired bifurcation curve will lie on the
two-dimensional surface (Fig. 3(a))
mapping parameter plane II {(c,/3):
< c < 4;
1/31 < into domains of various oscillatory modes,
is shown in Figure 5. The domains of chaoticity
adjoin to the parts of boundaries of 1-I,., formed
by the set of accumulation points. The diagram
depicts only one of them, which occupies the
largest region in the parameter plane. The domain
is shown in Figures 4, 5 by dark hatching and is
denoted as l-[Chao s. The remaining domains are
very narrow and hence, are not shown in the
diagram. There is a great number of windows with
deterministic dynamics within IlChaos, which begin
with saddle-node branching m-cycles. The inner
structure of such domains may be similar to that
for Ilk,./or be different. The domains, whose properties are different from the properties of fik,.j,
are denoted as Hswallw (see Fig. 4). The structure
of
and the bifurcations in this domain
were described in some detail in (Dmitriev et al.,
Ilfwallow
We find an equation for the bifurcation curve, by
setting X(c,/3, 1) equal to zero:
--8[#(Ct,/)](1/2) _+_ (1 )(1 +/)2
0.
The form of this curve is shown in Figure 3(b).
The remaining domains II3.2i-,1 i-2,3,..., as
well as ]-[2/-,1, i--3, 4,..., were built numerically.
Figure 4 shows the collection
1-[3,1
U3.2
i-1,1
U fik’2i-I
l-[k,1
i=1
k
i=1
5, 6, 7, 8, 10, 12
1994).
For any point P C Il with the exception of
P=(0,0) there is such (simply or not simply
connected) domain D, in the phase space of the
dynamic system (1), that if X0 c D,, then
lim
X(X0)]]- oc, X(Xo)
F(Xo).
In Figure 5 (cf. also Fig. 2) lii and M 2 denote
those domains of II (not hatched), where Eq. (9)
with any X0 is valid for all points P I)l t I)2.
As seen in Figures 4, 5, all sets FI,.j, with the
exception of wallw, have non-empty intersections with II,. Moreover 11i,l Ilq,p
(i, q 1,
i- q). This means, that there exist different stable
m-cycles, corresponding to IIi, and IIq,p for every
II
P
IIi, lf’qilq, p.
It is easy to see, that
and
IIk,2
U 1-Ik’2i-’,
i=1
2’
k
5, 6, 8
(9)
II3,1
Cl
AI-I
II5swallw
wallw
5,
b, 1-IiSwallw
i7 j.
BIFURCATIONS OF THE
FIGURE 3 (a) The 2D surface, on which the bifurcational curve
3-cycle.
Based on these results let us elaborate on the
bifurcations analysis, while moving along the
parameters in the domain (U(,.j)H,..) u fIChao in
more detail.
The analysis was made in the sections /3=0.3;
/3=0.68 and /3=0.5, changing the parameter c
over the range of
-
..-a(- 1)
2
< c < u, 1/31 < 1, u < 4, (u,
const)}.
The bifurcation diagrams calculated for these
values of / and the indicated variation of c are
N+
HI’NON MAP
209
for 3-cycle lies. (b) The curve of saddle-node bifurcation for
depicted in Figures 6(a), 7(a) and 8. It is easy to see
in these diagrams, that the complication of
oscillations happens through a sequence of
period-doubling bifurcations while continuously
increasing c. These sequences lead to the appearance of aperiodic motion with some critical value
of cx and then to the domain of chaoticity. The
ranges of variation in c in which chaotic oscillations are observed, are interrupted by small
intervals, where stable cycles exist (Figs. 6-8).
In the domain -(1/4)(fi/- 1) 2 < Oz < 1/1 < 1,
, < 4, (t,,/3 const) with variation in c many other
stable motions occur in saddle-node bifurcations,
such as 3-, 6-, 8-, 18-cycles and other motions with
,
Z.T. ZHUSUBALIYEV
210
et al.
H/0,1
H6,1
-1
-1
FIGURE 4 The collections of the sets ]-I3,1, Ik,2i-,,l,i
the chaoticity.
ot
1,2,... ,k
4
5, 6, 7,8, 10, 12 and I-[k, 2, k- 5, 6,8. Here IIChao is the domain
-1
FIGURE 5 Resulting picture for the division of H6non map parameter plane into domains of various oscillatory modes.
subsequent period-doubling bifurcations. That is
why the general pattern of cycles branching is
complicated significantly.
Let us follow the evolution of various stable
cycles, with variation in c. The results of
numerical calculations for /3=0.3 and /3=0.68
are presented in Tables I, II. In these tables the
second column contains the periodicity of a cycle,
the forth and the fifth columns are the range of c,
in which locally stable m-cycle exists. The last
column indicates the width of this range.
It is convenient to represent the data of the
tables in the form of a diagram, which was called branching pattern (Figs. 6(b) and 7(b)) in
BIFURCATIONS OF THE
HI’NON MAP
211
2.5
13=o.3
-1.5
13=0.68
-2.5
0.0
ot
0.65
1.4
(a)
0.98
ot
(a)
40
50
13=0.3
m
2
1.4
0
(b)
FIGURE 6 (a) Bifurcation diagram and (b) branching pattern
for 0 _< c _< 1.4 and/3
=0.68
0.3.
(g)
FIGURE 7 (a) Bifurcation diagram and (b) branching pattern
for 0.65 _< c < 0.98 and/3-0.68.
2.5
(Baushev and Zhusubaliyev, 1992; Baushev
et al., 1996). In these diagrams values of varying
parameters are plotted on the horizontal axis
and values of m are plotted on the vertical axis.
Further, we use designations as in (Baushev et al.,
1996). Vk,j. are the branches, where index k indicates from which k-cycle the branch begins. The
maximum value j indicates the number of observed branches, having the same k. All branches start
with k-cycles, which arise in a saddle-node bifurcations. Some Vk,j are determined on sets, that were
not shown in Figure 5, as their sizes in parameter plane are too small.
Xi
-2.5
0.7
1.074
ot
FIGURE 8 Bifurcation diagram for 0.7 <
/3=0.5.
c
_< 1.074 and
Z.T. ZHUSUBALIYEV et al.
212
TABLE
Vk,j
O i-1
m
Vl,1
V6,
V7,
V20,1
V2,1
2
4
8
16
32
64
6
12
7
14
28
56
20
21
OZ
2.k,
2
3
4
5
6
7
2
2
3
4
--0.1225
0.3675
0.9125
1.0258554050738
1.0511256620352
1.0565637581588
1.0577304656609
1.062372
1.0710703629071
1.2266174
1.2541834642429
1.2600151726701
1.2614289068416
1.067564
1.26887
2.k,
0.3675
0.9125
1.0258554050738
1.0511256620352
1.0565637581588
1.0577304656609
1.057956
1.0710703629071
1.0750124047736
1.2541834642429
1.2600151726701
1.2614289068416
1.2617
1.0677430954543
1.2692033433938
/
0.49
0.545
0.1133554050738
0.0252702569614
0.0054380961236
0.0011667075021
0.0002255343391
0.0086983629071
0.0039420418665
0.0275660642429
0.0058317084272
0.0014137341715
0.0002710931584
0.0001790954543
0.0003333433938
TABLE II
i-
O2.k,
V1,
V6,2
Vs,
V18,1
2
4
8
16
32
64
6
12
24
48
8
16
32
64
18
36
2
3
4
5
6
7
2
3
4
2
3
4
2
0.0768
0.808
0.9169381427467
0.9361479503607
0.9400330556502
0.9408680756308
0.70142261
0.7343659105724
0.7445017603884
0.7467731395103
0.70791248
0.713479366256
0.7158180118385
0.7163564950326
0.74696954
0.7476245768696
The range of existence of each branch Vk,.. is
presented in the form of intervals
Here the parameters OZ2ik,j, i= 1,2,... correspond to loss of stability of the 2 i- k-cycle and
soft occurrence of the 2ik-cycle. When increases,
the intervals /OZZik, j O2i+lk, j --OZZik, j Of the existence of the 2ik-cycle become narrower (see Tabs.
I, II), so that each branch has its limit value of
ck,j*, which corresponds to aperiodic motion. We
shall call these parameters accumulation points as
ozi2.k,
0.808
0.9169381427467
0.9361479503607
0.9400330556502
0.9408680756308
0.9410471449543
0.7343659105724
0.7445017603884
0.7467731395103
0.7472652867434
0.713479366256
0.7158180118385
0.7163564950326
0.7164739420036
0.7476245768696
0.7479207404767
/
0.7312
0.1089381427467
0.019209807614
0.0038851052895
0.0008350199806
0.0001790693235
0.0329433005724
0.010135849816
0.0022713791219
0.0004921472331
0.005566886256
0.0023386455825
0.0005384831941
0.000117446971
0.0006550368696
0.0002961636071
in (Baushev et al., 1996). The accumulation points
are shown in Figures 6(b) and 7(b) as vertical lines.
The beginning of the windows of periodicity in the
bifurcation diagrams coincides with the beginning
of the branches. Some of Vk,(k: 1) intersect with
V1,1. In those ranges of c, where V, [..J Vk,.-Cb,
k-/: 1, there exist both a stable 2 i- 1-cycle and stable
2 d- k-cycle, k _> 3; i, d_> 1. This means, that the
phase plane (x,x2) of the dynamic system (1)
comprises some basins of attraction of different
cycles. In this case the 2 i- 1-cycle or the 2 d- kcycle are chosen, depending on the initial
BIFURCATIONS OF THE
conditions. The character of division of the
iphase plane of system (1) into basins of attraction
of different cycles is shown in Figures 9-11. Figures 9(a) and (b) display basins of attraction for
the 1-cycle and the 3-cycle (P=(0.949;-0.744)).
HI’NON MAP
213
In Figure 10 we show basins of attraction for
the 1-cycle, the 18-cycle, and the 3-cycle,
(P=(1.0295;-0.8)), and in Figure 11 are basins
of attraction for the 2-cycle and the 8-cycle
(*’ (0.53;0.SS)).
X
-0,7
x,
(a)
FIGURE 9 (a), (b) Basins of attraction for the 1-cycle and the 3-cycle with
1,6
c
0.949,
fl
-0.744. (See Color Plate II.)
Z.T. ZHUSUBALIYEV
214
-2
et al.
FIGURE 10 Basins of attraction for the 1-cycle, the 18-cycle, and the 3-cycle with
FIGURE
Basins of attraction for the 2-cycle and the 8-cycle with c
The location of X,k, k-- 1,m, are marked by
dots in Figures 9-11. The non-hatched parts of
Figures 9 and 10 correspond to the collection of
initial conditions from which the trajectories
diverge to infinity (see(9)).
One can see from the comparison of the
branching pattern with the bifurcation diagrams
2
X
(--
1.0295,/3= -0.8. (See Color Plate III.)
0.53, /3= 0.88. (See Color Plate IV.)
that there are intervals within the limits of
existence of the V,I where periodic motions
coexist with chaotic oscillations. Here, the
system may chose a periodic motion or chaotic
oscillations, depending on the initial conditions. As an example Figure 12 gives the result
of plotting of the basins of attraction for the
FIGURE 12 Basins of attraction for the coexisting 6-cycle and chaotic motion with s =1 .7, fl = -0.03 . (See Color Plate V .)
a
(a)
+
1' 0,5
Swallow
A 5,1
Swallow
10,1
Fold
) 1,5
I
^ Swallow
I10,2
(U)
FIGURE 13 (a) The 2D surface illustrating the peculiarities of organization of the domains of existence for 5-cycle Itiallow and 10cycle ~SO,ill°" and Ilso211ow which are of a "swallow tail" kind . (b) Bifurcation curves bounding the domains H a11 o`" fl Mow and
nsozlloW
Note that both plots are not drawn to scale .
216
Z.T. ZHUSUBALIYEV
coexisting 6-cycle and chaotic motion (P=
(1.7;-0.03)).
Now let us consider the difference between the
properties of the set wSwaow and the set 11,...
Similarly to (Anishchenko, 1990), when analyzing
the model of the radiophysical generator with
inertial non-linearity, we introduce and consider
a three-dimensional space (,/3, x.(P)), in which
we depict the dependence x,:(P). Here, as in
(Anishchenko, 1990), by x..(P) we mean one of
component of the fixed point X:(P) of the
mapping X:=F’(X:). The locus, corresponding
to stable and unstable m-cycles, forms a twodimensional surface 6)m,.i in (c, ,x:(P)). The first
index in the designation of (3m,.. indicates the value
of m, for which the surface was built, and the
second index is introduced to distinguish (R),..,
having the same m. The plot of such a surface for
m= 5 and m= 10 is schematically shown (6)5,1
and 6)10,1, 6)10,2) in Figure 13(a).
Now we shall project an image of the surface
6),.. onto the plane of control parameters
(Fig. 13(b)). For the purpose of illustration, the
et al.
bifurcation curves in Figure 13 are not drawn to
scale. The domains of existence of the stable 5cycle and the stable 10-cycles are denoted as
Swallow
Swallow
Swallow
115,1 and H10,1 ,1110,2 respectively. The set
Swallow
1-[5,1 is limited_by the period-doubling bifur_c_tion curvves P,5 P 1,5 and the curves P+0,5’ P+1,5’ 1,5
(P,5, Pl,5 -are the lines of fold (Arnold, 1990) of
to combination of the stable
5-cycle and the
+ 5-cycle (the curve N+ ).
The curves P+1,5 and 1,5 are supported by the
(35,1), corresponding
unstable
Swallow
Fold
of codibifurcation point O Fld
1-I5,1
1,5 (01,5
mension 2 (the point of assembly (Arnold, 1990) of
Swallow
the projection 5,1), forming a sector in 115,1
where two stable 5-cycles and two unstable 5cycles exist. One of the unstable 5-cycles occurs
together with the stable 5-cycle on the boundary
p+0,5"
The picture below the curves P{5 and
Swallo
(Figs. 13,
repeats structure described for 115,1
and
14). The bifurcation curves
j- 1,...,2 i-1", i- 1,2,... accumulate, and there
FIGURE 14 The results of numerical calculations of the domains nSwllow k
P75-2’---’
5, 6, 7 8 and some others. (See Color Plate VI.)
BIFURCATIONS OF THE
HINON MAP
217
cz=1.53
o.14
O. 165
f
(a)
50
Vs2
ct=1.53
m
o.4
f3
FIGURE 15 (a) Bifurcation diagram for c=1.53 and 0.14_</3<0.165 and (b) branching pattern for c=1.53 and
0.145 _</3 _< 0.162. This section intersects that region of the domain 1-I5Swallw where two stable 5-cycle coexists.
exist transversal directions along which infinite series of period-doubling bifurcations take
place.
,
The projections of the two-dimensional surfaces
i= 2, 3,...)
lm,j, (m 5 2 i- j= 1,..., 2 ionto the place of control parameters have the
same peculiarities as 5, (Fig. 14). The domain
II5swallw consists of the collection of the sets
;
Swallow
1-[5.2i_1,j
j- 1,..., 2 i- 1., i- 1,2
I-5Swallw
/ (U
\j=lI5.2i-l,j;.
--Swallw
Figure 14 (see also Figs. 4, 5) shows the
collection of some I-[/wallw, which begin with
Z.T. ZHUSUBALIYEV
218
et al.
1.2
Xl
0=1.56
0.14
0.155
(a)
50’
.56
m
5
0.14
15
[
0.16
FIGURE 16 (a) Bifurcation diagram for a 1.56 and 0.14 _</3 _< 0.155 and (b) branching pattern for a 1.56 and 0.14 _</3 _< 0.16.
This section intersects that region of the domain II5swallw where two stable periodic, stable periodic and chaotic or two chaotic
motions can coexist.
different stable k-cycles
I-Iwallw
u
i=1
U I-Ik.2i_l,j
k
5, 6, 7, 8.
j=l
the parameters, as IlSwallow
The properties of
"k,conv
-iSwallow
adduced properto
similar
the
above
are
k,conv
ties of IIk,.. There is non-uniqueness of motions in
which li.e_ between
the domain l-[Swallow\I-ISwallow
"k
\’k,conv
the bifurcation curves
and Fj,5.2;_,, j1,...,2i-’; i= 1,2,... (Fig’sl
14).
When the boundaries of 1-[Swallw\llSwallw
"k
\l-k,conv are
+
and
intersected in the points of the curves Fj,5.2;_,,
+52;
3,
Let us denote the part of II2wallw, where a single
stable motion exists, which is uninterrupted along
BIFURCATIONS OF THE HINON MAP
.+
’j,5.2i_ ;j- 1,..., 2 i- 1., i- 1,2,
one can observe
hard transitions from one mode to another with
hysteresis as it is typical for such transitions
(Figs. 15 and 16). Figures 15 and 16 illustrate
only one of possible variants of change in the
character of the dynamics with variation in the
parameters. The branching patterns in other
sections may be more complicated in comparison
with those shown in Figures 15(b) and 16(b) (see
Figs. 17 and 18).
219
We realize that the content of the above analysis
is qualitative rather than quantitative. As it is seen
from Figure 14 the real picture of the properties of
wallw as a whole is
substantially more complicated than presented in this paper.
Finally, the case should be considered, when the
trajectory of deformation passes through oFold
"-’j,5.2i-1,
j- 1,..., 2 i- 1., i- 1,2,... The dependence of the
solutions of xc(P) (2) on the parameters for this
case is qualitatively shown in Figure 19, where
1-I
X
[3=-0.55ot+1.079
1.68
0.89356
FIGURE 17 Bifurcation diagram for/3= -0.55c+ 1.079 where 0.89356 _< c < 1.68.
1.5
X
-0.24
0.245
[3
FIGURE 18 Bifurcation diagram for
c= 1.53
and -0.24 _</3 < 0.245.
Z.T. ZHUSUBALIYEV
220
and reverse period-doubling bifurcations,
wallw
whereas in
hysteretic transitions are
possible.
3. While intersecting the boundaries of Hk,j., which
correspond to the parameters of hard occurrence of stable cycles, catastrophic transitions
from one stable motion to another or catastrophic chaotization are possible. But such
transitions are not hysteretic.
4. The sets IIk,j intersect non-emptily. Some of the
regions II,j. have intersections with the domains where chaotic oscillations are realized,
and that is why a great variety of bifurcation
transitions is possible.
IIc
S/
X
X
Xe
FIGURE 19 The qualitative plot of the variation of the stable
and unstable 5-cycles with the parameters when the trajectory
of deformation passes through the bifurcation point 0TM
1,5 of
codimension 2.
xS_ (P) and
et al.
xS+(P) correspond to the stable
x(P) corresponds to the unstable
55cycles, and
the
solutions
cycle. When approaching O Fld
1,5
xs_ (P), xSc+(P) and x(P) become closer and then
and form the
they combine in the point O Fld
1,5
single stable f-cycle, which is dependent on P
uninterruptedly. The solution x.(P) is nonrough at
the point O Fld
1,5 and, if we change the trajectory of
deformation in such a way that it does not pass
then x.(P) falls into two isolated
through O Fld
1,5
solutions.
3. CONCLUSIONS
1. In the present paper we have established the
domains of the modes of periodic and chaotic
oscillations in the plane of parameters using
numerical as well as analytical approaches.
Two types of the domains of cycles stability:
1-Ik,j, 1-I/wallw, k- 1,2,... were determined. The
structure of sets IIk,j, IIkswalw and their properties in whole have been studied.
2. It was shown that the motions, determined on
the sets IIk,j, depend smoothly on the parameters. While moving along the parameters
continuously within the limits of IIk,j, the
transition from some stable cycles to other
cycles occurs softly through a sequence of direct
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